Smoothed Particle Hydrodynamics Modelling of Bergy Bit and Offshore Structure Interactions Due to Large Waves

Abstract

This research utilised an open-sourced smoothed particle hydrodynamics (SPH) tool to model and predict the change in wave-induced forces and motions of a free-floating bergy bits approaching a fixed structure in regular waves. Simulation parameters, including particle resolution, fluid viscosity, initial wave condition and boundary treatments, are varied, and their effect on the load imparted to the bergy bit and the structure are investigated. The predicted motions are compared with previously published physical measurements for corresponding scenarios. Both predictions and measurements showed that, in regular waves, the surge motion slowed as the bergy bit approached the structure, and the heave motion increased. For wave loading on bergy bits, the agreement with the experimental data for the root mean square (RMS) force was within 2%. The pressure and velocity fields of the wave–structure–bergy bit interactions are discussed in light of the SPH predictions. This work confirms that the SPH model can accurately capture viscosity–dominated interactions, hydrodynamic damping, and eccentric impact like phenomena and predict both the impact and hydrodynamic loads due to a bergy bit drifting in waves towards a fixed offshore structure.

1. Introduction

In the presence of ocean waves, glacier ice fragments such as bergy bits and growlers impose high loads on ships and fixed or floating offshore structures due to direct impact and hydrodynamic interactions. A practical and efficient means to assess the interactions and facilitate the design of the offshore structures is to utilise well-validated numerical tools to accurately predict the impact and hydrodynamic loads due to the interactions. A significant amount of research has attempted to address this problem to date. Developing numerical techniques to accurately predict the impact and hydrodynamic loads is challenging due to the complex viscosity–dominated interactions, where negative wave drift force (against the direction of the propagation of waves), shadowing, changes in added mass, hydrodynamic damping, and eccentric impact like phenomena should be accordingly captured. The International Standard ISO 19906 [1] also emphasises the proximity hydrodynamic interaction effect with regard to ice–structure interactions.

Previous research has established that the impact loads resulting from a bergy bit are heavily influenced by the velocity at impact, which is intricately linked to the complex hydrodynamic interaction between the bergy bit and the structure [2]. Various experimental and numerical studies have observed a decrease in velocity prior to impact, along with the effects of reflected and standing waves, such as fluid cushioning, shadowing, and eccentric impact [3,4,5,6]. However, none of these studies have quantitatively assessed how these effects impact the final impact velocity.

The significance of the near-field or proximity effect has been highlighted for conventional fixed or floating offshore structures like gravity-based structures (GBS) or floating production storage and offloading units (FPSO), yet apart from [2], no research has been conducted on the hydrodynamic interaction with bergy bits in the very near-field region for these structures. While the far-field velocity of bergy bits has been extensively studied using potential flow models, investigations into near-field cases are relatively scarce. Sayeed [2] noted that most current studies overlook the hydrodynamic interaction immediately preceding the collision.

Hydrodynamic interaction between waves and multi-bodies poses significant challenges in advancing research. One prominent issue is accurately predicting resonant-free surface elevations in gaps between multi-bodies. Although non-viscous flow models are commonly employed to assess multi-body interactions, e.g., [7,8], these methods do not consider the viscosity effect, which can be crucial for near-field interactions.

Research on interactions between bergy bits and structures under calm water conditions has made significant strides in the past two decades. Gagnon and Wang [9] utilised the arbitrary Lagrangian–Eulerian (ALE) formulation within LS-Dyna (LSTC) to simulate interactions between flexible offshore platforms and flexible icebergs, including collisions. In a similar vein, Gagnon and Derradji-Aouat [10], as well as Song et al. [11], employed the ALE technique to model the impact of towed flexible bergy bits on rigid structural panels. The effectiveness of the ALE technique in modelling the fluid–structure interaction (FSI) involving deformable bodies and hydrodynamic effects has been demonstrated in these works. However, simulating interactions between wave-driven icebergs or bergy bits and offshore structures introduces additional complexities. There remains a gap in understanding how to effectively apply these techniques to model and simulate such intricate scenarios.

High-fidelity, finite volume method (FVM)-based CFD methods can combine wind, wave, current and structures in a single numerical domain. They offer promises for modelling the interactions between a bergy bit and a structure in a wave field at various proximities and captures the viscous dominant interactions. Bai et al. [12] was the first research group using RANS-based CFD codes to study ice floe drift in regular waves. Per the research, the linear flow model overestimated the surge and heave RAOs. However, the predictions from CFD analysis compared reasonably well to experimental data. Sayeed [2] utilised RANS-based commercial CFD code Flow3D to model various aspects of simplified bergy bits and fixed offshore structure interactions in multiple regular and irregular waves. The predicted highest and lowest wave loads on the bergy bit at different proximity distances from the offshore structure compared well with the corresponding measurements. Also, Sayeed’s [2] Flow3D model captured well the velocity changes at node and antinode locations in front of the structure. Somewhat related to the above studies, in more recent research, Huang and Thomas [13] used Flow3D to build a three-dimensional model to predict the movements of a circular ice floe drifted by regular waves in different wave conditions. The authors observed two noteworthy motion behaviours of the ice piece: overwash and scattering. The CFD model was validated against experiments and proven to be accurate in predicting the rigid body motion of the ice floe in the small regular waves; however, this technique may not be suitable for large waves where non-linear interactions and large drift motions of floating bodies are expected.

The majority of the existing literature on numerical methods utilising potential flow codes and CFD methods for predicting multi-body interactions in waves has been conducted in far-field conditions and small waves. The hydrodynamic interactions between floating bodies in close proximity, particularly in the presence of large waves, appear to be complicated and challenging due to the expected complex response. Knowledge gaps exist in understanding the response of a free-floating structure under the action of large waves. Classical CFD-based time-domain approaches and volume-discretised methods, such as the FVM and FEM, face difficulties in mesh generation and simulations for the highly deforming computational domain. Based on the evidence in the open literature, most numerical works are conducted with traditional RANS-based CFD solvers, and a few simplified studies are simulated using ALE approaches. The traditional Eulerian mesh-based approach provides satisfactory results but suffers from mesh distortion problems, especially during large drift motions. Even though overset mesh techniques handle this problem better than the moving mesh method, the final impact scenario cannot be modelled using such CFD methods. The smoothed particle hydrodynamics (SPH)-based approach, a meshless Lagrangian technique, is expected to handle the meshing entanglement issue better and simulate large waves and structure interactions in close proximity to and impact with other structures. That is why we explored the SPH application to our problem at hand. The main contribution of this study is to demonstrate the application of the SPH technique in a multibody wave–structure interaction study and validate the results with experimental data.

The objectives of the presented research include the following:

• Investigating the changes in wave loads on a simplified bergy bit at different proximities to a fixed structure using SPH modelling and comparing them with corresponding measurements;
• Investigating the changes in wave-induced motions of and wave loads on a free-floating bergy bit using SPH modelling in regular waves and comparing them with measurements;
• Investigating the changes in wave-induced motions of and wave loads on a free-floating bergy bit approaching a fixed structure using SPH modelling in regular waves and comparing them with measurements.

This study examines the motion characteristics and forces acting on a bergy bit under the influence of wave actions, both with and without the presence of a fixed offshore platform. Through SPH-based numerical simulations, this research provides valuable insights into how bergy bits interact with large waves under different conditions. The novelty of this work includes the implementation of the SPH technique and validation of the models for predicting the multi-body interactions in large waves. The authors are unaware of published studies of a floating body interacting with a fixed structure in large waves using fully non-linear SPH-based tools.

The numerical model utilised in this study employs a fully nonlinear viscous flow model based on weakly compressible smoothed particle hydrodynamics (SPH), which is implemented in the open-source SPH tool DualSPHysics (version 5.0.1). The methodology unfolds as follows: Section 2 offers a concise overview of the SPH method and its implementation. In Section 3, the accuracy of the numerical models is scrutinised through a comparison with experimental data from [2]. Section 4 presents a detailed analysis of the numerical results and discussions concerning a fully six-degree-of-freedom (6-DOF) 3D bergy bit model exposed to large regular waves. Finally, Section 5 encapsulates the research findings, evaluating the accuracy and efficiency of the SPH method and suggesting avenues for future investigation.

2. Numerical Methods

2.1. Smooth Particle Hydrodynamics (SPH) Method

SPH currently represents one of the most popular meshless Lagrangian particle models [14]. Over the last two decades, SPH methods have been applied to simulate a broad range of flow phenomena and fluid–structure interaction scenarios. Aristodemo et al. [15] provided an excellent summary of the applications of SPH techniques in different engineering fields. Per [15], the applications of SPH tools have been demonstrated through the modelling of highly non-linear flow phenomena such as the jets in water bodies [16], modelling of complex environments [17], sloshing phenomenon [18,19], water entry of solid bodies [20], impacts of flow on structures [21], and open channels with free-surface [22]. In the field of coastal and offshore engineering, the SPH approach has been applied to investigate wave propagation [23], wave generation and absorption processes [24], the interaction of regular or irregular waves with vertical structures [25], perforated breakwaters [26] porous breakwaters [27], floating objects [28,29] and wave energy devices [30,31], for example. SPH studies involving horizontal circular cylinders refer to their interaction with steady current flows [32,33]. Other relevant SPH-related research includes investigations on wave breaking and solitary waves [34], the tsunami impact force [35,36], oil spill modelling [37], moored floating structures [25,28,38,39], and hydroelastic problems [40,41].

The present research utilised the existing capabilities of the open-sourced SPH tool known as DualSPHysics. DualSPHysics follows the fundamentals of the SPH method, which is a Lagrangian and meshless method. In SPH, the numerical domain is discretised into a set of spherical particles that are nodal points where physical quantities (such as position, velocity, density and pressure) are computed as an interpolation of the values of the surrounding particles. The contribution of these neighbours is weighted using a kernel function (W) that measures that contribution starting from the initial particle spacing. The region of influence of the kernel is defined by the characteristic length, also known as the “smoothing length (h),” which is used to normalise the inter-particle distance. The kernel function is defined such that the contribution of particles beyond a cut-off distance (here, 2h) is not considered. A brief description of the SPH basics, the governing equations and the boundary conditions implemented and wave modelling techniques used in DualSPHysics is provided below.

2.1.1. SPH Basics

The mathematical fundamentals of SPH are based on integral interpolants. SPH is an interpolation method whereby any function can be expressed using the values of the different properties (velocity, density, pressure) of a set of particles. The basic principle of SPH is to approximate any function F (r) by the discrete integral interpolants using the smoothing kernel function W (r, h) as shown in Equation (1):

$$F\left(r_a\right)\approx \sum _{b}F\left(r_b\right){\displaystyle \frac{{\rho }_b}{m_b}}W\left(r_a-r_b,h\right)$$

(1)

where ρ_b and m_b are the density and mass, respectively, r_a − r_b is the distance between particles, and h determines the maximum distance between interacting particles. The summation is calculated over all b particles within the region where the kernel is defined. The performance of an SPH model depends heavily on the choice of the smoothing kernel. Kernels are expressed as a function of the non-dimensional distance between particles (q), given by q = r/h, where r is the distance between any two given particles, a and b, and the parameter h (the smoothing length) controls the size of the area around particle a in which neighbouring particles are considered. Within DualSPHysics, the Quintic [42] kernel from the fifth-order Wendland kernel family (Equation (2)) is often recommended:

$$W\left(r,h\right)={\alpha }_D{\left(1-{\displaystyle \frac{q}{w}}\right)}^4\left(2q+1\right)\hspace{1em}\hspace{1em}\hspace{1em}0\le q\le 2$$

(2)

where q = (r_a − r_b)/h, and α_D is equal to 7/4πh² in 2D and 21/16πh³ in 3D. Following the definition of q, the smoothing radius is 2h.

2.1.2. Governing Formulations

The mass and momentum conservation laws (Navier–Stokes equations) that govern the dynamics of fluid motion are given in Lagrangian form; see Equations (3) and (4):

$${\displaystyle \frac{D\rho }{Dt}}=-\rho \nabla ·\mathit{v}$$

(3)

$$\begin{gathered} {\displaystyle \frac{D\nu }{Dt}}=-{\displaystyle \frac{1}{\rho }}\nabla \mathrm{p}+\mathrm{\Gamma }+f \\ {\displaystyle \frac{D\nu }{Dt}}=-{\displaystyle \frac{1}{\rho }}\nabla \mathrm{p}+\mathsf{\nu }{\nabla }^2\mathit{v}+g \end{gathered}$$

(4)

where D denotes the total or material derivative, t is the time, v is the velocity, p is the pressure, Γ represents the dissipation terms, f represents accelerations due to external forces, such as gravitational acceleration, ν is the kinematic viscosity, and g is the gravity.

The mass conservation property of SPH can be written as dm/dt = 0, where the mass is conserved exactly within a Lagrangian particle, resulting in a density change due to the volumetric change of the term at the right-hand side of Equation (3). Employing the standard SPH interpolation approximation [43,44] to discretise the conservation equations and incorporating the laminar viscous stresses and sub-particle scale (SPS) to represent the effects of turbulence lead to the SPH forms of the Navier–Stokes equations at point a with position r_a becoming the following [43,44]:

$${\displaystyle \frac{{D\rho }_a}{Dt}}=\sum _b{m}_b\left(v_a-v_b\right)·{\nabla }_a{\mathrm{W}}_{ab}+2\delta h{c}_0\sum _b{m}_b\left({\displaystyle \frac{{\rho }_a}{{\rho }_b}}-1\right){\displaystyle \frac{r_{ab}}{r_{ab}^2+{\eta }^2}}·{\nabla }_a{W}_{ab}$$

(5)

$$\begin{gathered} {\displaystyle \frac{{Dv}_a}{Dt}}=-\sum _b{m}_b\left({\displaystyle \frac{p_a}{{\rho }_a^2}}+{\displaystyle \frac{p_b}{{\rho }_b^2}}\right){\nabla }_a{\mathrm{W}}_{ab} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sum _b{m}_b\left({\displaystyle \frac{4\nu {\mathit{r}}_{ab}·{\nabla }_a{W}_{ab}}{\left({\rho }_a+{\rho }_b\right)\left(r_{ab}^2+{\eta }^2\right)}}\right){\mathbf{v}}_{ab} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sum _b{m}_b\left({\displaystyle \frac{{\tau }_a}{{\rho }_a^2}}+{\displaystyle \frac{{\tau }_b}{{\rho }_b^2}}\right){\nabla }_a{\mathrm{W}}_{ab}+g \end{gathered}$$

(6)

where $r_{ab}=\left|{\mathit{r}}_{ab}\right|,{\mathit{r}}_{ab}=r_a-r_b,{\mathit{v}}_{ab}=v_a-v_b$ and ${\nabla }_a{W}_{ab}={\nabla }_{a}W(r_a-r_b$, h). The second term on the right-hand side of Equation (5) is delta-SPH [45], which is also known as the density diffusion term (DDT). This term is diffusive and helps to reduce the numerical noise from the density fluctuations and to stabilise the pressure field. The variable $\delta$ is a free parameter and needs to be attributed to a suitable value. However, $\delta =0.1$ is recommended in DualSPHysics, and works well for most applications [46,47]. The second term on the right-hand side of Equation (6) is the discrete form of the viscous stress [48], whereas the third term is the sub-particle-scale (SPS) term, where τ is the SPS stress tensor [23]. DualSPHysics also offers the artificial viscosity scheme proposed by Monaghan [49], which is a common method within fluid simulation using SPH due primarily to its simplicity. This scheme is used in the current investigation.

In DualSPHysics, the SPH formulation of the above two equations treats the fluid as weakly compressible, and Tait’s equation of state is used to couple the two equations to determine fluid pressure based on particle density [50]. Following [43], the pressure and density are coupled by means of an equation of state (EOS), allowing for the weak compressibility of the fluid based on the numerical speed of sound, which follows the expression in Equation (7):

$$p={\displaystyle \frac{{\rho }_0{c}_0^2}{\gamma }}\left[{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{\gamma }-1\right]$$

(7)

where ${\rho }_0$ is the reference density, $c_0=c\left({\rho }_0\right)={\left.\sqrt{\left({\displaystyle \frac{\partial p}{\partial \rho }}\right)}\right|}_{{\rho }_0}$ is the numerical speed of sound at reference density, ${\rho }_0$ = 1000 kg/m³, p is the pressure, and $\gamma$ is an empirical constant that depends on the fluid (e.g., $\gamma =7$ for water). According to Equation (7), the compressibility of the fluid depends on c₀, in such a way that for a high-enough sound celerity, the fluid is virtually incompressible. However, the value of c₀ in the model should not be the actual speed of sound, as the stability region is defined by Equation (5). The numerical speed of sound, c₀, is chosen based on a typical length scale and timescale of the domain, which allows for much larger time steps within the explicit time integration than would be possible with a physical speed of sound. With c₀ = 10‖v‖_max, where ‖v‖_max = √(gh_o), with h_o being the initial fluid height in the domain, a variation in density of up to about 1% generally occurs. This should, however, be monitored as there are special cases where 1% may be exceeded, and compressibility may cease to be ‘weak.’ In such cases, the speed of sound should be increased with an inevitable decrease in time step and an increase in computational time [51].

In DualSPHysics, two explicit time integration schemes, namely the Verlet time integration scheme and the Symplectic time integration scheme, are implemented to solve the governing equations (short forms for the brevity):

$${\displaystyle \frac{D{v}_a}{Dt}}=F_a;{\displaystyle \frac{D{\rho }_a}{Dt}}=R_a;{\displaystyle \frac{D{r}_a}{Dt}}=v_a$$

(8)

The time integration is bounded by the Courant–Friedrich–Levy (CFL) condition necessary in explicit time integration schemes to limit the numerical domain to the physical domain of dependence [51].

$$\begin{gathered} \mathsf{\Delta }{t}_f=\underset{a}{\min }\left(\sqrt{{\displaystyle \frac{h}{\left|\frac{d{v}_a}{dt}\right|}}}\right); \\ \mathsf{\Delta }{t}_{cv}=\underset{a}{\min }{\displaystyle \frac{h}{c_s+\underset{b}{\max }\frac{\left|h{v}_a.r_a\right|}{r_{ab}^2+{ɳ}^2}}} \\ \mathsf{\Delta }t=C_{CFL}min\left(\mathsf{\Delta }{t}_f,\mathsf{\Delta }{t}_{cv}\right); \end{gathered}$$

(9)

where |dv_a/dt| is the magnitude of particle acceleration. The variable time step is chosen as the minimum of Δt_f and Δt_cv, and is bounded by the Courant number, C_CFL, usually in the range of 0.1 to 0.2.

Further details of the basics of SPH formulations and their implementations in DualSPHysics are presented in [47,51].

2.1.3. Boundary Conditions

In DualSPHysics, the boundary is described by a set of particles that are considered a separate set from fluid particles. The software currently provides functionality for solid impermeable and periodic open boundaries. Methods to allow boundary particles to be moved according to fixed forcing functions are also available.

The dynamic boundary condition (DBC) is the default method provided by DualSPHysics to model solid boundaries [47,52]. This method sees boundary particles that satisfy the same equations as fluid particles; however, their positions and velocities are not updated. Instead, they remain either fixed in position or move according to an imposed/assigned motion function (i.e., moving objects such as gates, wave-makers or floating objects). This approach is not based on a rigorous physical derivation and is known to suffer from accuracy issues. Nevertheless, this approach is easy to implement, computationally efficient, and naturally handles general 3D shapes without the need to know any geometric information about the boundary.

The stability of this method relies on the length of the time step taken being suitably short in order to handle the highest present velocity of any fluid particles currently interacting with boundary particles, and is, therefore, an important point when considering how the variable time step is calculated. All the SPH simulations carried out in this work used this boundary condition to define the boundaries of the structure and the numerical tank.

Within DualSPHysics, it is possible to define a pre-imposed movement for a set of boundary particles. Various predefined movement functions are available, as is the ability to assign a time-dependent input file containing kinematic detail. These boundary particles behave as a DBC described above; however, rather than being fixed, they move independently of the forces currently acting upon them. This provides the ability to define complex simulation scenarios (i.e., a wave-making paddle) as the boundaries influence the fluid particles appropriately as they move [52].

2.1.4. Regular Wave Simulations

DuaSPHysics is capable of modelling complex wave phenomena by employing advanced numerical techniques within the smoothed particle hydrodynamics (SPH) framework. Within DualSPHysics, defining a pre-defined movement for a set of boundary particles is possible. Several such movement functions and the capability for assigning a time-dependent input file containing kinematic detail are available in the software package. These boundary particles behave as a dynamic boundary condition (DBC) described in [52,53,54], except that they move independently of the acting forces. This attribute allows the user to define complex simulation scenarios (i.e., a wave-making paddle/piston) as the boundaries influence the fluid particles appropriately as they move.

Piston-type long-crested wave generation schemes with second-order correction for bound long waves (sub-harmonics) available in DualSPHysics are used to generate monochromatic (regular) waves in the current research. Also, a passive wave absorption technique available in DualSPHysics is used, which enables generations of long-time series of waves in relatively short domains with negligible wave reflection. Passive wave absorbers are usually required to dampen the wave energy and reduce the reflection exerted by the boundary of the model domain. Further details of these wave generation and passive wave absorption techniques can be found in [52,53,54].

2.1.5. SPH–DEM Coupling

The impact of the bergy bits and the fixed offshore structure (solid–solid interactions) is modelled using the discrete element method (DEM) available within DualSPHysics, which is based on [55,56]. Canelas et al. [56] introduced a unified discretisation of rigid solids and fluids, allowing for resolved simulations of fluid–solid phases within the meshless SPH framework. The process included the numerical solution attained by SPH and a variation of the DEM, distributed contact discrete element method (DCDEM) discretisation, which is achieved by directly considering solid–solid and solid–fluid interactions. Canelas et al. [56] generalised the coupling of the DEM and SPH methodologies for resolved simulations, allowing for state-of-the-art contact mechanics theories to be used in arbitrary geometries, while fluid-to-solid and vice versa momentum transfers are accurately described over a broad range of scales. The general concept is to compute the forces acting on a fluid–solid particle pair using the SPH formulation and solid–solid interactions via the DEM, retaining the same explicit integrator and also the DualSPHysics meshless framework. Per [55], given adequate non-linear contact models, a wide range of material behaviours can be modelled using three basic parameters: Poisson’s ratio, Young’s modulus and the dynamic friction coefficient.

Forces arise whenever a particle of a solid object interacts with another. In a solid–solid collision, the contact force is decomposed into Fn and Ft, normal and tangential components. Both of these forces include viscous dissipation effects. This is because two colliding bodies undergo a deformation that is somewhere between perfectly inelastic and perfectly elastic, usually quantified by the normal restitution coefficient; see Equation (10):

$$e_n=-{\displaystyle \frac{\left.u_n\right|t=t_n}{\left.u_n\right|t=0}},\mathrm{e}\in \left[\mathrm{0,1}\right]$$

(10)

where t = t_n is the instant at the end of collision and t = 0 is the instant immediately before. Forces are further decomposed into a repulsion force, F_r, arising from the elastic deformation of the material and a damping force, F_d, for the viscous dissipation of energy during the deformation. Other dissipative mechanisms, such as plastic deformation and the emission of elastic waves excited from impact, are not considered. The elastic waves are always present but carry very little energy and are usually disregarded. Plastic deformation is not considered directly as its effects can, to some extent, be included in the viscous dissipation terms. Figure 1 generally illustrates the proposed viscoelastic DEM mechanism between two interacting particles [55].

Every solid–solid interaction is conceptualised as a system of springs and dampers. A general expression for the normal force in such a viscoelastic model is given in Equation (11):

$$F_{n,ij}=F_n^r+F_n^d=K_{n,ij}{\mathsf{\delta }}_{ij}^{p1}{e}_{ij}-{\mathsf{\gamma }}_{n,ij}{\mathsf{\delta }}_{ij}^{p2}{e}_{ij},$$

(11)

where k_n,ij is the normal stiffness constant of pair ij, δ_ij = max(0,(di + dj)/2 − |r_ij|) is the particle overlap, e_ij is the unit vector between the two mass centres and γ_n,ij is the normal damping constant. The stiffness and damping coefficients are given by Equation (9), where eⁿ_ij is the unit vector between the centres of particles i and j. The mechanism is linear for p1 = p2 = 1, corresponding to a simple damped harmonic resonator. An analytical solution shows that this leads to a constant normal restitution coefficient independent of the impact velocity [55].

$$\begin{gathered} K_{n,ij}={\displaystyle \frac{4}{3}}{E}^{*}\sqrt{R^{*}} \\ {\mathsf{\gamma }}_{n,ij}={\displaystyle \frac{{loge}_{ij}}{\sqrt{{\pi }^2+{log}^2{e}_{ij}}}} \end{gathered}$$

(12)

The restitution coefficient, e_ij, is taken as the average of the two material coefficients, which is the only calibration parameter of the model. It is not a physical parameter, but since the current method does not account for internal deformations and other energy losses during contact, the user can change this parameter freely to control the dissipation of each contact. The reduced radius and reduced elasticity are given by Equation (13):

$${\mathrm{R}}^{*}={\left({\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}\right)}^{-1};\mathrm{E}={\left({\displaystyle \frac{1-{\nu }_{p1}^2}{E_1}}+{\displaystyle \frac{1-{\nu }_{p2}^2}{E_2}}\right)}^{-1}$$

(13)

Ri is the particle radius, E_i is the Young modulus, and ν_p is the Poisson coefficient of material i.

Regarding tangential contacts, friction is modelled using the same model as that of normal contact and is given by Equation (14):

$$F_{t,ij}=F_t^r+F_t^d=K_{t,ij}{\mathsf{\delta }}_{ij}^t{\mathrm{e}}_{ij}^t-{\mathsf{\gamma }}_{t,ij}{\mathsf{\delta }}_{ij}^t\dot{{\mathsf{\delta }}_{ij}}{\mathrm{e}}_{ij}^t,$$

(14)

where the stiffness and damping constants are derived using Equation (15), so as to insure internal consistency of the time scales between normal and tangential components:

$$\begin{gathered} K_{t,ij}={\displaystyle \frac{2}{7}}{K}_{n,ij} \\ {\mathsf{\gamma }}_{t,ij}={\displaystyle \frac{2}{7}}{\mathsf{\gamma }}_{n,ij} \end{gathered}$$

(15)

This mechanism models the static and dynamic friction mechanisms using a penalty method. The body does not statically stick at the point of contact, but is constrained by the spring–damper system. This force must be bounded above by the Coulomb friction law, modified with a sigmoidal function in order to make it continuous around the origin regarding the tangential velocity, as stated in Equation (16):

$$F_{t,ij}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\mu }_{ij}{F}_{n,ij}tanh\left(8{\mathsf{\delta }}_{ij}^t\right){\mathrm{e}}_{ij}^t,F_{t,ij}\right)$$

(16)

where μ_ij is the friction coefficient at the contact of object i and object j and is simply taken as the average of the two friction coefficients of the distinct materials.

The drawbacks of the model are common to all general DEM formulations: their explicit nature may impose very small stability regions, extended frictional contacts are either computationally expensive or inaccurately reproduced and low-resolution simulations may lead to artificial geometry effects due to the underlying spherical shape of the particles. Further details of the coupling of DEM with the SPH method implemented in DualSPHysics and its applications are presented in [47,55,56].

3. Modelling and Validation

3.1. Numerical Wave Tank

A numerical wave tank (NWT) was modelled by incorporating wave generation using Piston-type second-order wave generation techniques [48] and the wave absorption/damping phenomena using a passive absorption technique. Firstly, a three-dimensional particle block was generated. Then, the target waves were generated with the bergy bit placed at the centre of the NWT without any other structure placed in the domain. This scenario investigated the free drifting of the bergy bits in waves. The interactions between the bergy bit and offshore platform were modelled by introducing a fixed vertical structure to the simulation domain at a particular distance from the wave maker. During the wave modelling, the second-order waves were created by correcting the first-order wave generated at the wave maker. The correction technique included compensation for both parasitic long waves and displacement long waves. This correction method implemented in DualSPHysics is based on the solution for the wave maker’s control signal described in [54]. The length of the wave absorption zone was close to one-half of the largest wavelength. The centre of the bergy bit was placed at least two wavelengths away from the wave maker to minimise any wave reflection from the bergy bit reaching the wave inlet. The width of the NWT was set to be at least four times that of the fixed structure to minimise the boundary effects.

All waves were generated in the transitional water depths, primarily to have a manageable number of particles in terms of computational efforts required. Figure 2 presents the sketch and layout of the numerical wave tank with relaxation/damping zones. A solid/rigid spherical bergy bit with a 0.305 m diameter and 0.875 specific gravity was modelled, and a hexagonal cylinder measuring 0.5 m in diameter was used as the fixed offshore structure (see Figure 3). The bergy bit’s centre of gravity was 190 mm below its top.

3.2. Simulation Cases

Several SPH simulation cases were prepared to investigate the free surface wave interactions with a floating bergy bit with and without the presence of a fixed structure in different wave conditions. Three regular waves with different lengths and steepness were selected for the study.

Table 1. The regular waves used for the model simulation and validation.

	H (m)	T (s)	λ/DS	λ/H	T (Full-Scale)
RW1	0.1	1.39	6	30	9.83
RW2	0.0875	1.06	3.5	20	7.50
RW3	0.1125	1.2	4.5	20	8.49

shows the incident wave properties studied in this research for both the simulation scale and full scale. Simulations were carried out for three scenarios: firstly, for the bergy bit fixed in space and at different distances away from the vertical offshore structure; secondly, for the bergy bit drifting in waves alone; and, lastly, for the bergy bit drifting towards the fixed offshore structure.

Table 2. Simulation conditions for bergy bit force validations.

ID	Waves	Di/Ds	λ/DS	D (m)	D/Di
C1	RW2	0.6	3.5	0.12	0.4
C2	RW2	0.6	3.5	0.61	2.0
C3	RW2	0.6	3.5	1.25	4.1
C4	RW2	0.6	3.5	2.0	6.6
C5	RW2	0.6	3.5	4.0	Inf

shows the simulation setup carried out with the bergy bit at different separation distances from the fixed structure.

Table 3. Simulation conditions for bergy bit motion validations.

ID	Structure	Waves	λ/DS	Initial D (m)	No. of Repeats
D1	NO	RW1	6	NA	3
D2	NO	RW2	3.5	NA	2
D3	NO	RW3	4.5	NA	1
E1	YES	RW1	6.0	2.0	2
E2	YES	RW2	3.5	2.0	2
E3	YES	RW3	4.5	2.0	1

shows the simulation setup completed with the bergy bit moving freely with and without the presence of the offshore structure. The choice of the specific operating conditions in our current study emerged from the conditions experimentally tested by Sayeed [2]. Waves were also constrained by the facility tank water depth and wave generation capability. The scaling factor was assumed to be 50, similar to that of the baseline experiments conducted by Sayeed [2]. The waves were selected to cover full-scale wave periods of 7 to 10 s and heights of 4 to 6 m. At this scale, the bergy bit represents a body measuring 15 m in diameter, and the offshore platform represents a gravity-based structure with a 25 m characteristic length at the water surface. Overall, this scenario represents a generic condition prevalent in the sub-arctic areas, off the coast of Newfoundland and Labrador Canada.

In the above Tables, Di is the bergy bit diameter, Ds is the vertical cylindrical structure diameter, λ is the wavelength, and D is the distance between the bergy bit and the vertical structure. As shown in Table 3, the separation distance was based on multiples of the bergy bit diameter. Four separation distances were covered for the 0.305 m diameter, Di, of the sphere. This resulted in a ratio of the bergy bit diameter to the structure diameter (DS) of Di/DS = 0.6.

Table 4. Detailed parameters for bergy bit motions in waves simulations.

SPH Simulation Parameters	Values
Inter-particle distance, dp	0.0075 m
Coefficient to calculate the smoothing length (h = coefh × sqrt(3 × dp2) in 3D	1
Viscosity formulation	Artificial
Viscosity value	0.01
Reference density for the fluid, Rho0 (minimum and maximum)	1000 (700 and 1300) m3/kg
Polytropic constant for water used in the state equation, Gamma	7
Coefficient to multiply speed system	30
Cfl number, coefficient to multiply dt	0.2
Restitution coefficient	0.0
Step algorithm	Symplectic
Interaction kernel	Spline
Density diffusion term	Molteni
SPH simulation parameters	Values
Inter-particle distance, dp	0.0075 m

provides different computational parameters and choices used in all DualSPHysics simulations. The selection of the computation parameters and values is based on the work completed by Islam et al. [57], which offers a detailed sensitivity study for the interparticle distance, smoothing length coefficient and viscosity modelling type and value. In the simulations, the inter-particle distance value was used so that at least 15 particles spanned over the wave heights. Regarding modelling the viscosity effect, the artificial viscosity scheme, proposed by Monaghan [43,49], is a common method within fluid simulation using SPH due primarily to its simplicity. An artificial viscosity value of 0.01 gives the best results in validating wave flumes to study wave propagation and wave loadings exerted onto coastal structures [25]; hence, the same value is used in all current simulations. The restitution coefficient is related to the average of the coefficients of the two materials (that collide). It is not a physical parameter, but since the DEM does not account for internal deformations and other energy losses during contact, this parameter offers the flexibility to control the dissipation of each contact. More details of the definitions of other parameters and the rationale for the selected values are provided in [46].

Table 5. Final SPH simulations, particle size and computation time.

Simulation Scenarios	Total Particle Size (Millions)	Computation Time
Wave only	8.6	12.5 days
Fixed bergy bit and fixed structure in waves	8.8	13 days
Drifting (free-floating) bergy bit in waves	14.6	25 days
Drifting bergy bit (free-floating) and fixed structure in waves	15.2–46.5	25 days

provides the total number of particles and the computation time needed for different simulation scenarios completed in this research. The simulations were completed at multiple particle resolutions for the drifting bergy bit and the fixed structure in wave scenarios, resulting in total particle numbers ranging between 15.2 million and 46.5 million. All simulations were carried out for at least 60 s of physical time. At least ten steady-state wave cycles were selected to analyse the data. The majority of the simulations were completed using HPC Intel(R) Xeon(R) Gold 5118 CPU @ 2.30GHz, 48 CPU core, 128 GB memory, NVIDIA Quadro P2000 (4 GB dedicated GPU memory and 1000 GPU core, NVIDIA: Santa Clara, CA, USA). The high-resolution (>20 million particles) simulation cases were carried out using NVIDIA DGX system [58].

The data predicted included the time series of forces in the x, y, and z directions (FX, FY, and FZ, respectively) of the bergy bit and the fixed structure. The surface elevations were also predicted at different locations of interest in the NWT. The data analysis included evaluating the basic statistical values such as the average, maximum, minimum, standard deviation and root mean square values of a selected time segment (steady cycles) of the time series data. The initial transition periods of the time series data of bergy bit motions and forces were not included in estimating the statistical properties. Each predicted data series contained at least ten cycles of responses. The wave-only loads (FX and FZ) on the bergy bit were derived from the corresponding time series by removing any static offset, as applicable. Further analysis was carried out for all cyclic time-series data (wave elevation; bergy bit heave motion and forces (FX and FZ)) to determine the magnitudes and periods. The peaks and troughs of a cyclic time series are obtained using “Zero Crossing Analysis” or ZCA, also known as crossing analysis [59]. The maximums of the motions and forces were calculated by taking the average of their peak values. Note that the physical measurements carried out by Sayeed [2] did not include the forces on the fixed structure and the moving bergy bit. Hence, the SPH predictions of these forces could not be validated.

4. Results and Discussion

4.1. SPH Predictions of the Wave Field

A thorough study has been carried out to ensure accurate predictions of the generated regular waves at the locations of interest in the NWT. The sensitivity study included varying the inter-particle distance (or the particle size), smoothing length, and the width of the NWT. It was found that an inter-particle distance of <=0.01 m (approximately 1/30 of the bergy bit diameter or 1/15 of the wave height) and a smoothing length of <=1.2 were sufficient to reproduce waves with an accuracy comparable to that of the dispersion theoretical wave for regular deep-water waves. A width of the NWT >= 4 * bergy bit diameter was deemed sufficient. Figure 4, Figure 5 and Figure 6 show the predicted surface elevation at multiple locations (around the initial location of the bergy bit) for the three regular waves, showing the accuracy of the model predictions compared with the deep-water dispersion theoretical elevation.

Table 6. Accuracy of the SPH model-generated waves.

	% Difference of H (m)	% Difference of T (s)
RW1	1.11	0.52
RW2	1.25	0.75
RW3	0.93	0.35

compares the average wave height and periods of the three regular waves between the predicted and the target theoretical waves. The SPH model reproduces waves with an accuracy comparable to that of the theoretical waves. The predicted wave heights and periods were within 1.5% and 1.0%, respectively, of the corresponding deep-water theoretical waves in all locations checked. This confirms the accuracy and consistency of the wave generations predicted by the SPH tool along and across the NWT.

4.2. Wave Forces on Fixed Bergy Bits at Different Proximities

These simulation cases included positioning the bergy bit fixed at different proximity distances from the fixed offshore structure (see Table 3). Figure 7, Figure 8 and Figure 9 show the NWT predictions by the SPH model, the positioning of the two structures, and the velocity distribution at the free surface.

The SPH model predicted the time series of FX, FY and FZ forces for the bergy bit with Di/Ds = 0.6 located at D/Di = 0.4, 2.0 and 4.1 in wave λ/Ds = 3.5 (RW2), which are shown in Figure 10, Figure 11 and Figure 12. These results are typical of all the raw data collected during the SPH predictions. The predicted FY values were negligible for all fixed bergy bit cases as the waves were symmetrical about the centreline of the NWT; hence, they were not analysed as part of the study. It is observed that both FX and FZ magnitudes and profiles change as the proximity of the sphere to the structure is reduced. The same phenomenon was observed during the experiments carried out by Sayeed [2]. Also, SPH simulations capture non-linear FZ forces very well, indicating a shallower draft of the bergy bit. Note that the first 20 s of the time series data are not presented (and not considered for further analysis) due to their transitory nature.

Figure 13 shows the SPH-predicted mean drift force, FX, for the bergy bit Di/Ds = 0.6 at different separation distances and for the RW2 wave. Generally, the predicted values follow a similar trend to those recorded in the measurements (yellow triangles in the figure). A negative FX value is predicted at D/Di = 0.4 for the wave under investigation. This may be attributed to the negative drift force acting on the body against the direction of the as the body gets close to the structure. The physical significance of this phenomenon is that the negative force, when it exists, may help the bergy bit avoid impacting the structure. The drift force was positive and at its maximum as the bergy bit was placed at D/Di = 2.0. As the bergy bit was moved further away, the force tended to reduce but stayed positive for the rest of the distances. The SPH model predicts the proximity effect on the drift force on the bergy bit well in terms of magnitude and trends. Note the small mean values compared with the first-order wave loads of RMS FX forces acting on the body. This small mean drift force is important in determining the drift speed, the direction of the bergy bit movements and the latter’s potential to impact a structure.

Figure 14 shows the mean vertical force, FZ, for the same bergy bit at different locations and for the same wave as that above. The mean FZs are all positive at all separation distances, which means that FZ acts downwards for all cases with or without the presence of the structure, which is intuitive as the majority portion of the sphere is submerged. Also, the mean FZ increases in magnitude as the sphere gets close to the structure. The trends are well captured by the SPH model.

Table 7. Comparison of RMS of FX and FX between measurements and SPH predictions for sphere Di/Ds = 0.6 and RW1 (cases B1–B5).

D/Di	RMS FX (N)			RMS FZ (N)
	Basin [2]	SPH	% Diff	Basin [2]	SPH	% Diff
Inf	15.48	15.32	−1.0%	6.72	7.59	11.5%
0.40	10.67	10.55	−1.1%	10.01	9.96	−0.5%
2.00	19.93	19.48	−2.3%	5.02	5.74	12.5%
4.10	14.64	15.06	2.8%	7.65	9.02	15.2%
6.70	16.61	16.97	2.1%	6.31	6.33	0.3%

shows the RMS FX and FZ comparisons for the bergy bit in wave λ/Ds = 3.5 (RW2). The results match pretty well, showing the highest and lowest forces at different locations from the structure. The predicted FX and FZ were primarily within 3% and 15% of the corresponding measurements. The low accuracy of the FZ predictions may be attributable to the differences in setup for the bergy bits. The SPH simulations did not include the attached rod used during experiments. This may explain the slight difference in RMS FX and the larger difference in FZ. Another factor may be the minor differences in wave heights generated between the experimental facility and the numerical wave tank. The SPH results indicate that the viscous forces are negligible compared with the pressure forces for the test cases.

4.3. Wave-Induced Motions of Bergy Bit

This section investigates the change in wave-induced motions of free-floating bergy bits without other structures. In addition to the motions, the velocity profiles of the drifting bergy bit are studied and evaluated to determine how that velocity changes before impact. Additional analysis was carried out to determine the possible correlations between motions and the forces predicted in the previous section for when the bergy bit is in proximity to the fixed structure.

The 6-DOF motions of the bergy bit are predicted at the centre of gravity. The origin (0,0,0) of the motion is considered at the initial position of the bergy bit in the x-direction. Figure 15, Figure 16 and Figure 17 present the time series for surge and heave motions from the measurements and the SPH predictions for sphere Di/Ds = 0.6 in wave λ/Ds = 6.0 (RW1) and λ/Ds = 4.5 (RW3). In the figures, SPH01, 02 and 03 mean that the simulations were carried out at the same NWT configurations with slight differences in the release time of the bergy bit.

In both basin measurements and SPH model predictions, per Figure 15 and Figure 16, both low- and high-frequency motions are contained in the x displacement time series data. The high-frequency motions are the oscillatory motions due to the wave frequency, and the low-frequency motions are due to the second-order wave drift forces. Overall, the SPH model predicted the path, magnitude and oscillation trends captured during the measurements. Some differences were expected since the initial conditions of the sphere release in the experiments and SPH simulations were not the same. Sayeed [2] observed that the trajectory differs even for similar conditions because of the difference in the initial release condition (whether the bergy bit was released in the crest or trough of the wave). This effect is more dominant for small bodies in large waves. The heave motions showed high-frequency oscillations similar to the wave frequency and predicted the heaving amplitude in the same order of magnitude as the corresponding measurements, as shown in Figure 17. Overall, the predicted oscillatory surge and heave motion agree well with the corresponding measurements. The SPH-predicted results also captured well the non-linearity in the heave motions in wave λ/Ds = 6. This was also expected as non-linear heave forces were observed in earlier sections, as shown in Figure 12. Because of the spherical shape of the bergy bit, the roll, pitch and yawing motions were not present (near-zero).

4.4. Bergy Bit Impact on Offshore Platform

This section investigates the change in wave-induced motions and the resulting forces of free-floating bergy bits approaching a stationary structure. Unlike the previous analysis, the structure face located in the x direction was considered the origin for the motions. The initial/starting position of the bergy bit was 2 m upstream of the face of the structure. During the simulations, the bergy bit was held fixed for 10 s to let a few wave cycles pass by and ensure that steady-state reflections were developed from the structure. A total of five (5) simulations were carried out, see

Table 8. Summary of SPH-predicted bergy bit and structure collisions.

ID	Waves	Motion Constraints	No. of Collisions
SPH01	RW1	Sway	1.00
SPH02	RW1	None	None
SPH03	RW2	None	3.00
SPH04	RW2	None	2.00
SPH05	RW3	None	None

. Figure 18, Figure 19 and Figure 20 show the time series of the surge, sway and heave displacements of the bergy bit for three different wave conditions. The heave motions of the bergy bit were significantly increased when the structure was present, particularly at a certain proximity. This is evident when Figure 17 and Figure 20 are compared. The increase in the heave motion was accompanied by a decrease in the surge motion for these two cases.

The simulations revealed that the sphere in waves λ/Ds = 6 (RW1) and λ/Ds = 3.5 (RW2) went and hit the structure, whereas the sphere in wave λ/Ds = 4.5 (RW3) went by the structure and did not hit it. These motion profiles also indicate the change in velocity throughout the trajectory, as shown in Figure 21. The figure demonstrates that the SPH model captured the velocity changes at node and antinode locations of the wave field. It is to be noted that only the SPH01 simulation had the sway motions disabled, but the bergy bit had all motions activated for the other simulations.

Next, we look into the forces on the bergy bit as well as on the structure as the bergy bit approached the structure and, in some cases, collided with it. Table 8 summarises the simulations completed for this scenario and the nature of the bergy bit impacting the structure. Figure 22, Figure 23 and Figure 24 show the FX, FY and FZ on the bergy bit due to the wave drifting towards the structure. The impact forces (in FX and FZ) are clearly visible in the time trace and identified in the figure legends. The impact is more visible in the FX than in the FZ time traces. The total FX at the impact appears to be up to 400% of the FX due to wave drift.

In both measurements and SPH, predictions of the heave motion, velocity, and FZ showed non-linear behaviour for most wave cases. The increase in heave motion and velocity corresponds to a reduction in surge motion and velocity at the bergy bit about to impact the structure. Sayeed [2] discussed non-linear heave motions and forces of spherical bergy bit models. In general, it is hypothesised that the very small freeboard of the floating sphere representing the floating bergy bit is responsible for the non-linearity in the vertical motion, velocity and force. During drifting in waves, the water is observed to quickly wash over the top surface of the bergy bit, completely submerging the berg (or model), which is depicted in the non-linear behaviour. During the wave–bergy bit interactions, the wake formed over the spherical body will rotate around it. The oscillatory back and forth sweeps for the sphere may also be responsible for the non-linearity.

Figure 25 shows the FX force on the vertical structure due to the wave actions. The impact force on the structure due to the bergy bit colliding with the structure is evident in the time series data. The FX on the structure at impact appears to be up to 20% of the FX due to wave drift. Further investigation is required to quantify the impact force and validate the quality of these force predictions. Further refinement of the particle resolutions (a finer time step) may be necessary to capture the exact impact force.

Overall, the results presented in this paper show satisfactory agreement with experimental model test data. Reasonable discrepancies are expected when simulating highly distorted breaking waves and very close proximity hydrodynamic interactions. The discrepancies between the SPH predictions and corresponding measurements are within an acceptable range from an engineering analysis point of view. The outcome of any SPH-based model depends on interparticle distances, smoothing length, kernel functions, numerical schemes, and parameter choices, which ultimately depend on the computational resources available. Considering the available computational resources and previous experience, the authors used the parameters and numerical schemes that were the most suitable to produce the desired wave field. To improve the accuracy of the predictions, the waves were calibrated with analytical solutions before implementing the more complicated wave–structure interaction modelling. The observed discrepancies may have also resulted from the uncertainties in the measurements, which need further investigation.

After investigating the wave forces and wave-induced motions in regular waves, the next logical step is to examine the phenomenon in irregular waves. This study may reveal whether or not the observations found in regular waves will still hold in irregular seas and whether or not linear superposition theory may be applicable to identify this phenomenon in irregular waves. The impact phenomena of the single- or multi-bergy bit interactions also need further studies with fixed and floating offshore platforms, which is the primary objective of the research group. The present study offers a basis for such advanced studies.

With traditional CFD, the above problem can be tackled with separate CFD and FEM methods (one-way coupling) or the complex coupling of the fluid–structure interaction, FSI (two-way coupling). The present work demonstrated that the SPH technique will facilitate the achievement of the overall research objective without much complexity in terms of simulation setup with a high level of accuracy. Even computational resources required for SPH simulations are less intensive than fully coupled FSI techniques. The present work demonstrated that the SPH techniques have the potential to solve complex engineering problems related to multi-body interactions in realistic ocean conditions, which will help in the design and safe and efficient operations of offshore energy applications.

5. Conclusions

An open-sourced SPH tool was used for the first time to investigate the change in wave-induced motions and forces of a free-floating bergy bit without and approaching a fixed structure in multiple regular waves. The predicted forces and motions are compared with previously published physical measurements for corresponding scenarios. Simulations were carried out for three scenarios: firstly, for the bergy bit fixed in space and at different distances away from a simplified vertical offshore structure; secondly, for the bergy bit drifting in waves alone; and, lastly, for the bergy bit drifting towards the fixed offshore structure. We arrive at the following conclusions:

The SPH-predicted forces on the static bergy bit at different distances from the vertical structure compared well with the corresponding measurements. The predicted root mean squared FX and FZ values were within 3% and 15% of the corresponding measurements. The low accuracy of the FZ predictions may be attributable to the differences in the bergy bit modelling and wave generation between the numerical and physical techniques.

The SPH model also predicted the motion characteristics of the drifting bergy bit well in terms of the path, magnitude and oscillation, which shows similar trends to those captured during the measurements. The comparison is considered qualitative and expected to have some differences since the initial conditions of the bergy bit release in experiments and SPH simulations are different.

Both predictions and measurements showed that, in a regular wave field, as the bergy bit approached the structure, the surge motion slowed, and at the same time, the heave motion increased. The bergy bit impact was more noticeable in the FX than in the FZ time traces. The total FX at the impact was up to 400% of the force due to wave drift. There were no corresponding measurements for comparison.

The research demonstrated that the SPH model can simulate and capture the effects of wave reflection and proximity in modelling the hydrodynamic forces and resulting motions for a small bergy bit approaching a fixed structure. The next step of this research will be to develop the SPH model further to accurately model a realistic irregular 3D wave field, the drifting of free-floating realistic bergy bits approaching complex offshore platforms (fixed and floating) and the possible impact scenarios.